\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 124, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2011/124\hfil Non-existence of limit cycles]
{Non-existence of limit cycles via inverse integrating factors}
\author[L. Laura-Guarachi, O. Osuna, G. Villase\~nor-Aguilar
\hfil EJDE-2011/124\hfilneg]
{Leonardo Laura-Guarachi, Osvaldo Osuna,
Gabriel Villase\~nor-Aguilar} % in alphabetical order
\address{
Instituto de F\'{\i}sica y Matem\'aticas,
Universidad Michoacana, Edif. C-3, Cd. Universitaria,
C.P. 58040. Morelia, Mich., M\'exico}
\email[L. Laura-Guarachi]{leonardo@ifm.umich.mx}
\email[O. Osuna]{osvaldo@ifm.umich.mx}
\email[G. Villase\~nor-Aguilar]{gabriel@ifm.umich.mx}
\thanks{Submitted September 1, 2011. Published September 27, 2011.}
\subjclass[2000]{34C05, 34C07, 34C25}
\keywords{Algebraic set; integrating factors;
inverse integrating factors; \hfill\break\indent limit cycles}
\begin{abstract}
It is known that if a planar differential systems has an inverse
integrating factor, then all the limit cycles contained in the domain
of definition of the inverse integrating factor are contained in
the zero set of this function. Using this fact we give some
criteria to rule out the existence of limit cycles.
We also present some applications and examples that illustrate our
results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\section{Introduction and statement of results}
Many problems in qualitative theory of differential equations in
the plane are related to limit cycles; this fact motivates their
study. We consider the system of differential equations
\begin{equation}\label{sys1}
\begin{gathered}
\dot{x}_1 = f_1(x_1,x_2) \\
\dot{x}_2 = f_2(x_1,x_2),
\end{gathered}
\end{equation}
where $f_i:U\subseteq \mathbb{R}^2 \to \mathbb{R}$,
$1\leq i\leq 2$ are functions of class $C^1$ and $U$ is a simply
connected open set. Consider the vector field
$F:=f_1\frac{\partial }{\partial
x_1}+f_2\frac{\partial }{\partial x_2}$, then system
\eqref{sys1} can be rewritten in the form
\begin{equation}\label{sys2}
\dot{x} = F(x), \quad x:=(x_{1},x_{2}) \in U.
\end{equation}
Its divergence is
$\operatorname{div}(F):=\frac{\partial f_1}{\partial x_1}
+\frac{\partial f_2}{\partial x_2}$.
\begin{definition} \label{def1} \rm
A function $\vartheta: U\subset \mathbb{R}$,
$\vartheta \in C^1(U, \mathbb{R})$, is said to be an inverse
integrating factor of \eqref{sys1} if it is not locally null
and satisfies the partial differential equation
\begin{equation}\label{sys3}
f_1\frac{\partial \vartheta}{\partial x_1}
+f_2 \frac{\partial \vartheta}{\partial x_2}
=\big( \frac{\partial f_1}{\partial x_1}
+\frac{\partial f_2}{\partial x_2} \big)\vartheta.
\end{equation}
\end{definition}
In short notation, an inverse integrating factor is a solution
of the equation $F\vartheta=\operatorname{div}(F)\vartheta$.
It is well known that inverse integrating factor is an important
tool in the qualitative study of differential equations,
but their determination is a difficult problem (see \cite{GG}
and references therein). In particular, in \cite{GJV} has been
established that are also a very useful tool for investigation
of limit cycles.
The aim of this article is to use inverse integrating factors
to produce in a systematic way, criteria for non-existence
of limit cycles in planar differential equations.
In particular, we obtain an alternative proof to the nonexistence
of limit cycles for homogeneous polynomial equations.
We give some examples to illustrate applications of these results.
We rewrite \eqref{sys1} in its Pfaffian form
\begin{equation}\label{sys9}
\omega:=-f_2(x_1, x_2)dx_1+f_1(x_1, x_2)dx_2=0, \quad (x_1, x_2) \in U.
\end{equation}
Note that the above equation is just the differential equation
of the orbits of system \eqref{sys1}. Recall that an integrating
factor for $\omega=0$ is a $C^1$ function $\mu:U\to \mathbb{R}$,
which makes $\mu \omega$ an exact form. In the case that $U$
is simply connected; this is equivalent to
\begin{equation}\label{sys10}
\frac{\partial (-\mu f_2)}{\partial x_2}
=\frac{\partial (\mu f_1)}{\partial x_1}
\end{equation}
It is clear that $\mu$ is an integrating factor for \eqref{sys9}
if and only if, $\vartheta=\frac{1}{\mu}$ is an inverse
integrating factor of \eqref{sys1}, in the appropriate domain.
Before establishing our results we recall the following result.
\begin{theorem}[{\cite[Theorem 9]{GJV}}] \label{thm1}
Let $\vartheta: U\to \mathbb{R}$ be an inverse integrating factor
of \eqref{sys1}. If $\gamma \subset U$ is a limit cycle of
\eqref{sys1}, then $\gamma$ is contained in the set
$\vartheta^{-1}(0):=\{(x_1, x_2) \in U : \vartheta(x_1, x_2)=0 \}$.
\end{theorem}
Recall that it is possible to impose certain conditions on
\eqref{sys10}, to determine special cases of integrating factors.
Our first result is an observation that these techniques can
be adapted to exclude existence of limit cycles.
We start with the following result.
\begin{proposition} \label{prop1}
Let $U$ be a simply connected open set.
Suppose a vector field
$$
F=f_1\frac{\partial }{\partial x_1}+f_2\frac{\partial }
{\partial x_2}\in C^1(U, \mathbb{R}^2).
$$
If any of the following two conditions holds,
then \eqref{sys1} does not have limit cycles in $U$:
\begin{itemize}
\item[(i)] The function
$\alpha_i:=\operatorname{div}(F)/f_i$ depends only on $x_{i}$,
for some $i\in \{1, 2\}$ and is continuous;
\item[(ii)] The function
$\beta:=\operatorname{div}(F)/\big(f_1x_2+f_2x_1\big)$ depends on
$z:=x_{1}x_2$ and is continuous.
\end{itemize}
\end{proposition}
\begin{proof}
We consider the case (i) with $\alpha_1$ depending only on $x_1$.
We seek an inverse integrating factor, using the associated equation
$$
f_1\frac{\partial \vartheta}{\partial x_1}+f_2
\frac{\partial \vartheta}{\partial x_2}
=\big( \frac{\partial f_1}{\partial x_1}
+\frac{\partial f_2}{\partial x_2} \big) \vartheta.
$$
Assume that $\vartheta$ depends only on $x_1$.
Thus the previous equation reduces to
$$
f_{1}\frac{\partial \vartheta}{\partial x_1}
=\big( \frac{\partial f_1}{\partial x_1}
+\frac{\partial f_2}{\partial x_2} \big) \vartheta,
$$
which is rewritten as
$$
\frac{\partial \log \vartheta}{\partial x_1}
=\frac{1}{f_{1}}\big( \frac{\partial f_1}{\partial x_1}
+\frac{\partial f_2}{\partial x_2} \big)=\alpha_1.
$$
From our hypothesis $\vartheta
=\exp (\int^{x_1} \alpha _{1}(s) ds )$ is an inverse
integrating factor and $\vartheta^{-1}(0)=\emptyset$,
therefore by Theorem \ref{thm1} system \eqref{sys1} has no limit
cycles. The proof is complete.
\end{proof}
\begin{example} \label{examp1} \rm
Consider the system
\begin {gather*}
\dot{x_{1}}= -x_2+ax_1^2+bx_2^3\cos(x_2),\\
\dot{x_{2}}= x_1.
\end {gather*}
We have that $\frac{\operatorname{div}(F)}{f_2}=2a$
is a function of $x_2$ so by Proposition \ref{prop1}(i).
Then the system contains no limit cycles.
\end{example}
\begin{example} \label{examp2} \rm
Consider the system
\begin {gather*}
\dot{x_{1}}= 2x_1x_2,\\
\dot{x_{2}}= x_1^3x_2^2-x_2^2-1.
\end {gather*}
We have that $\frac{\operatorname{div}(F)}{f_1}=x_1^2$,
by Proposition \ref{prop1}, this system contains no limit cycles.
\end{example}
We also have the following immediate result
(well known in the literature \cite[page 18]{Y}).
\begin{corollary} \label{coro0}
Let $F$ be a $C^1$ vector field on $U$. If $\operatorname{div}(F)=0$,
then \eqref{sys1} does not have limit cycles in $U$.
\end{corollary}
Now we use Proposition \ref{prop1} to study some special systems.
Consider the equation
\begin{equation} \label{sys6}
\begin{gathered}
\dot{x}_1=r_{1}(x_1)r_{2}(x_2), \\
\dot{x}_2=s_{1}(x_1).
\end{gathered}
\end{equation}
We establish the following result.
\begin{corollary} \label{coro1}
If $r_{1}(x_1)>0 (<0)$, then \eqref{sys6} does not have limit
cycles in $U$.
\end{corollary}
\begin{proof}
Indeed, the expression
$$
\frac{\operatorname{div}(F)}{f_1}=\frac{r'_1(x_1)}{r_1(x_1)},
$$
is continuous and depends only on $x_{1}$, hence the result
follows from Proposition \ref{prop1}(i).
\end{proof}
\begin{example} \label{examp3} \rm
Consider the system
\begin{gather*}
\dot{x_{1}}= (2+\sin(x_1))(x_2^3-x_2^2+x_2),\\
\dot{x_{2}}= x_1^4+5x_1.
\end{gather*}
It contains no limit cycles.
\end{example}
Recall that the phase portrait of differential equation
is essentially unchanged if we multiply the vector field by
a nonzero function.
\begin{lemma} \label{lem1}
Suppose that system \eqref{sys1} has a limit cycle $\alpha$
and $B: U\to \mathbb{R}$ is a positive (negative) real valued function.
Then $\alpha$ is a limit cycle of the system
\begin{equation}
\begin{gathered}
\dot{x}_1 = Bf_1, \\
\dot{x}_2 = Bf_2.
\end{gathered}
\end{equation}
\end{lemma}
Now using the above lemma, we obtain slightly general versions of
our results.
\begin{proposition} \label{prop2}
Let $U$ be a simply connected open set. Suppose that
$B: U \to \mathbb{R}$ is a $C ^ 1$ positive (negative)
function such that
$\operatorname{div}(BF)/Bf_i$ depends only on $x_{i}$, for some
$i\in \{1, 2 \}$ and is continuous.
Then \eqref{sys1} does not have limit cycles in $U$.
\end{proposition}
In particular from the preceding proposition or from
Corollary \ref{coro0}, we have the following result.
\begin{corollary} \label{coro3}
Let $U$ be a simply connected open set. Suppose that
$B: U \to \mathbb{R}$ is a $C ^ 1$ positive (negative)
function such that $\operatorname{div}(BF)=0$,
then \eqref{sys1} does not have limit cycles in $U$.
\end{corollary}
\section{Polynomial vector fields}
In this section we are mainly interested in studying polynomial
vector fields. We start presenting some basic concepts.
Let $\mathbb{R}[x_1, x_2 ]$ be the polynomial ring over
$\mathbb{R}$ in two variables. Given $f\in \mathbb{R}[x_1, x_2]$,
define its zero set by
$$
V(f):=\{(x_1, x_2) \in \mathbb{R}^2 : f(x)=0 \}.
$$
If $S\subset \mathbb{R}[x_1, x_2]$, we let $V(S)$ be the set
of common zeros
$$
V(S)=\cap_{f \in S} V(f).
$$
A set of this form is called algebraic, in particular $V(f)$
is known as algebraic curve.
\begin{lemma} \label{lem2}
Let $P \in \mathbb{R}[x_1, x_2 ]$ be a non-zero homogeneous
polynomial, then $V(P)$ contains no subset homeomorphic
to $\mathbb{S}^1$.
\end{lemma}
\begin{proof}
If $P:=c\neq 0$, then $V(P)=\emptyset$ and the result is valid,
so consider $P$ a homogeneous polynomial of degree $\geq 1$,
then note that $0\in V(P)$.
Suppose $V(P)$ contains a subset $\alpha$ homeomorphic to
$\mathbb{S}^1$. If it happens that $0 \in \operatorname{int}(\alpha)$
(the region bounded by $\alpha$), then $ P\equiv 0$ which is
a contradiction.
On the other hand, if we have $0 \notin \operatorname{int}(\alpha)$,
then we would have the cone $C(\alpha):=\{\lambda x : \lambda \geq 0,
\; x\in \alpha \}\subset V(P)$ which is a contradiction
to B\'ezout's theorem \cite{Fu}, because there are
lines without common components with $V(P)$, but an infinite
number of points of intersection. This completes the proof.
\end{proof}
Based on this lemma we can prove the following result.
\begin{theorem} \label{thm2}
If a non-zero homogeneous polynomial is an inverse integrating
factor of the differential equation \eqref{sys1}, then it has
no limit cycles.
\end{theorem}
\begin{proof}
Let $\vartheta$ be an inverse integrating factor of system
\eqref{sys1} and is homogeneous polynomial. Suppose the system
\eqref{sys1} has a limit cycle $\alpha$, then $\alpha \subset
\vartheta^{-1}(0)$ which contradicts Lemma \ref{lem2}. This concludes the
proof.
\end{proof}
We have an alternative proof of the following result,
which is proven in \cite{SP}.
\begin{corollary} \label{coro4}
If $f_1, f_2$ are homogeneous polynomials of same degree,
then \eqref{sys1} has no limit cycles.
\end{corollary}
\begin{proof}
It is easy to check that
$$
\vartheta(x_1, x_2):=x_1f_2(x_1, x_2)-x_2f_1(x_1, x_2),
$$
is an inverse integrating factor of \eqref{sys1}. It is clear that
$\vartheta$ is a homogeneous polynomial,
the result follows from Theorem \ref{thm2}.
\end{proof}
\begin{example} \label{examp4} \rm
Consider the system
\begin {gather*}
\dot{x_{1}}= 3x_2^5-x_1^3x_2^2+6x_1x_2^4,\\
\dot{x_{2}}= x_1^2x_2^3-2x_1^5.
\end {gather*}
This is a homogeneous vector field, by the above corollary, it
contains no limit cycles.
\end{example}
In particular, we have the following well known result.
\begin{corollary} \label{coro5}
The linear differential equation
\begin{equation}
\begin{gathered}
\dot{x}_1 = ax_1+bx_2, \\
\dot{x}_2 = cx_1+dx_2,
\end{gathered}
\end{equation}
contains no limit cycles.
\end{corollary}
Another application of Theorem \ref{thm2} is based on one of
the main results in \cite{CGaG}.
\begin{theorem}[\cite{CGaG}] \label{thm3}
Consider the polynomial system
\begin{equation} \label{sys15}
\begin{gathered}
\dot{x}_1=P_n(x_1, x_2)+x_1A_{d-1}(x_1, x_2), \\
\dot{x}_2=Q_n(x_1, x_2)+x_2A_{d-1}(x_1, x_2),
\end{gathered}
\end{equation}
where $P_n, Q_n, A_{d-1}\in \mathbb{R}[x_1, x_2 ]$ are homogeneous
and their degrees satisfy $d>n \geq 1$.
Assume that $H(x_1, x_2)$ is a $p$-degree homogeneous first integral of the
system
\begin{equation} \label{sys16}
\begin{gathered}
\dot{x}_1=P_n(x_1, x_2), \\
\dot{x}_2=Q_n(x_1, x_2).
\end{gathered}
\end{equation}
Then, the function
\begin{equation} \label{sys17}
\vartheta(x_1, x_2):=(x_2Q_n(x_1, x_2)-x_1P_n(x_1, x_2))
H(x_1, x_2)^{(d-n)/p}
\end{equation}
is an inverse integrating factor of \eqref{sys15}.
\end{theorem}
Now combining Theorem \ref{thm2} and \ref{thm3}, we have
the following consequence.
\begin{corollary} \label{coro6}
Under the hypotheses of Theorem \ref{thm3},
if $\frac{d-n}{p}\in \mathbb{N}$, then \eqref{sys15} is free
of limit cycles.
\end{corollary}
\begin{proof}
From the assumptions in Theorem \ref{thm3} and
$(d-n)/p\in \mathbb{N}$, it follows that $H(x_1, x_2)^{(d-n)/p}$
is homogeneous; therefore,
$$
\vartheta(x_1, x_2):=(x_2Q_n(x_1, x_2)-x_1P_n(x_1, x_2))
H(x_1, x_2)^{(d-n)/p}
$$
is a homogeneous polynomial, so the result follows from
Theorem \ref{thm2}.
\end{proof}
Now we consider the differential equation
\begin{equation} \label{sys20}
\begin{gathered}
\dot{x}_1=r_{1}(x_1)r_{2}(x_2), \\
\dot{x}_2=s_{1}(x_1)s_2(x_2).
\end{gathered}
\end{equation}
\begin{proposition} \label{prop3}
If $r_1$ and $s_2$ are polynomial functions then \eqref{sys20}
has no limit cycles.
\end{proposition}
\begin{proof}
A calculation gives that the function
$$
\vartheta(x_1, x_2):=r_1(x_1)s_2(x_2)
$$
is an inverse integrating factor of system \eqref{sys20}.
Now $V(\vartheta)=V(r_1)\cup V(s_2)$. Taking the factorization
in irreducible polynomials, one has that
$V(r_1)$ consists of a finite union of vertical lines,
similarly $V(s_2)$ is a finite union of horizontal lines.
So every subset of $V(\vartheta)$ homeomorphic to $\mathbb{S}^1$
must contain points in $V(r_1)\cap V(s_2)$; i.e., critical points
therefore can not be limit cycles.
\end{proof}
\begin{example} \label{examp5} \rm
Consider the predator-prey equation
\begin {gather*}
\dot{x_{1}}= x_1(a-bx_2)\\
\dot{x_{2}}= (cx_1-d)x_2,
\end {gather*}
where $a, b, c, d$ are positive constants.
By Proposition \ref{prop3} this system contains no limit cycles.
\end{example}
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\end{thebibliography}
\end{document}